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\begin{document}

\begin{center}
{\large{\bf Astro 201 -- Radiative Processes -- Problem Set 10}}

{\large{\bf Due in class.}}

\end{center}

\vspace{0.1in}
\ni
Readings: RL chapter 4.1, 4.9; paper by Carilli et al.~on class website;
however much you like of the classic paper by Scheuer
and Williams also on website.

\ni NOTE FOR ALL PROBLEMS: In this class, we are not going to stress
over the $\sin \alpha$ term in any expression involving
synchrotron emission, nor are we going to worry about
other factors of order unity, like gamma ($\Gamma$) functions.
If you see a $\sin \alpha$ or gamma function in your
travels, just set it equal to 1.

\ni Work in the same spirit that we have been working throughout this
class: order-of-magnitude. At the same time, try to hang onto
the $4\pi$'s.

\ni
{\large{\bf Problem 1.}} Leaving on a Jet Frame\footnote{Former BAD grad Julia Kregenow named this problem.}

\ni Two-sided jets travelling at close to the speed of light often appear brighter on one end than the other.

\ni Consider a bipolar (two-sided), symmetric jet of
material moving at BULK velocity $v$
and having BULK Lorentz factor $\gamma$. The jet axis points at an
angle $\theta$ relative to the Earth. One side
of the jet is directed towards us, and the other side
is directed away.

\ni The jet is composed of electrons
(possibly also positrons; the situation is unclear)
that radiate synchrotron emission. In the
rest frame of the jet, the electrons are still
gyrating relativistically, and in that same rest
frame, they emit
a specific intensity $I_{\nu} \propto \nu^{\alpha}$
between frequencies $\nu_1$ and $\nu_2$.
For this problem, take $\alpha < 0$.

\ni Recall from the reading that $I_{\nu}/\nu^3$ is a relativistic
invariant.

\ni (a) Sketch the specific intensity versus frequency of
the jet that points toward an observer on Earth,
as seen by that observer.

\ni Overlay on this sketch the specific intensity
of the counter-jet (again, as seen by the Earth-bound
observer).

\ni Overlay also the specific intensity of either jet
or counter-jet as measured in its rest frame.
Compare in particular the rest-frame jet
spectrum with the counter-jet spectrum observed at Earth.

\ni Annotate your sketch with scales
and indices, expressed symbolically in terms
of the variables given above.

\ni (b) By what (symbolic) factor
is the forward jet brighter than the counter-jet in specific intensity
at a fixed frequency as measured on Earth?

%\ni (Comment on astronomy jargon:
%the asymmetry in brightness is a result of ``Doppler boosting,''
%not of ``relativistic beaming'' or ``relativistic aberration.'')

\ni (c) The dynamic contrast in optical surface brightness
between jet and counter-jet in M87 exceeds 450.
If $\alpha = -1/2$, what are the constraints on $\gamma$, $v$, and
$\theta$?

\vspace{0.1in}
\ni
{\large{\bf Problem 2.}} In Space, No One Can Hear You Scream

\ni Take a look at Figure 3, Table 3, and Figure 4
of Carilli et al.'s paper on the radio galaxy Cygnus A.

\ni In Figures 4a and 4b, we can distinguish a break
in the spectral index at about $\nu \approx 10^4$ MHz.
The break is interpreted to represent the onset of synchrotron losses,
as we studied in a previous problem. The
corresponding electron energy spectrum will also be broken.

\ni In either regime, provided we don't look at frequencies
too low (lower than $\nu \approx 10^{2.75}$ MHz),
the emission arises from optically thin material.

\ni In Figure 3, one arcsecond equals one kpc.

\ni Part (d) is independent of the other parts.

\ni (a) Estimate the minimum magnetic field strength
in ``Hot Spot A'' using Burbidge's minimum energy argument
and using Figure 3, Table 3, and/or Figure 4, or any
other raw data from Carilli et al.'s paper.
Express in Gauss.

\ni You will have to decide whether to use the spectrum
below the break, above the break, or both. We want
a {\it minimum} estimate of the magnetic field.

\ni Interpret the unit of ``(beam)$^{-1}$'' (per beam)
to mean $[\pi (2.25 \, \rm{arcseconds})^2]^{-1}$.
This is the angular area of the radio interferometer's beam,
or footprint on the sky. When Carilli et al. cite ``4.5 arcseconds
resolution,'' I interpret the 4.5 arcseconds to be equal
to the FWHM (full-width-half-maximum) of their beam.

\ni (b) Estimate the minimum total pressure---magnetic plus material
(electron)---in hot spot A. Express in cgs units.
Compare to the pressure of air around you.

\ni (c) Estimate the total energy (magnetic plus material)
in hot spot A in ergs.

\ni OPTIONAL (d) Estimate an age
of the hot spot given the observation of the spectral
break. Compare to the dynamical time of the jet (the time
it took the jet to grow to its present size).

\vspace{0.1in}
\ni
{\large{\bf Problem 3.}} Galactic Synchrotron Emission

\ni The {\it brightness temperature} of synchrotron emission
in the Galactic plane is measured to be

\begin{equation}
T_{b} = 250 \left( \frac{\nu}{480 {\rm MHz}} \right)^{-2.8} \K \,,
\end{equation}

\ni valid for 8 GHz $> \nu > 480$ MHz. This emission arises from cosmic ray
electrons gyrating in the Galactic magnetic field of strength
$B \sim 3 {\mu}$G. Take the size of the emitting region to be $\sim$10 kpc.

\ni (Aside from being a nuisance foreground for CMB researchers, Galactic
synchrotron emission is thought to probe the supernova rate in the Galaxy,
since cosmic ray electrons and protons at the relevant energies are thought
to be accelerated in supernova shock waves. In addition,
a famous ``FIR-radio correlation''
is observed to exist for normal spiral galaxies; the radio traces
electrons accelerated by supernovae,
while the far-infrared (FIR) emission traces
warm dust grains. Where there are supernovae, there is active star formation;
where there is star formation, there are dust grains warmed by the ISRF.
Or so goes the traditional interpretation of the FIR-radio correlation.)

\ni (a) Provide an approximate expression for
the differential energy spectrum of cosmic ray electrons.
Express in units of particles $\m^{-2} \sr^{-1} \s^{-1} \GeV^{-1}$.
Indicate the approximate range of energies ($E_{\rm min}$ and $E_{\rm max}$)
for which your expression is valid.

\ni OPTIONAL (b) Compare your answer to the differential energy spectrum
of cosmic ray protons, as given in the Reader. The units
of the plot were accidentally chopped off; they are the
same as those specified in part (a). By what factor do
cosmic ray protons outnumber cosmic ray electrons?

\ni (c) Estimate the gyro-radii of cosmic ray electrons and of cosmic
ray protons at $E_{\rm min}$ and $E_{\rm max}$. Compare to the size
of the (baryonic disk of the) Galaxy, 10 kpc. From this comparison
you should be able to understand why cosmic ray astronomy is so difficult.

\ni OPTIONAL (d) Why do we ignore the contribution from cosmic ray protons to the
Galactic synchrotron emission at the above frequencies $\nu$?
Provide a quantitative answer.

\end{document}

\vspace{0.1in}
\ni
{\large{\bf Problem 6.}} Brown Dwarf's B

\ni Edo Berger and company detected
an unprecedented radio flare from brown dwarf LP944-20.
They detected a (spatially unresolved)
flux of $F_{\nu 1} = 0.5$ mJy (milliJansky) at $\nu_1 = 4.9$ GHz.
and a flux of 
$F_{\nu 2} = 2$ mJy at $\nu_2 = 8.5$ GHz.

\ni The emission is interpreted as synchrotron radiation
because (1) there is no way a cool ($T \approx 3000\K$)
brown dwarf could produce such radio power as
a thermal blackbody, and (2) it is polarized (and thermal
blackbody emission is not). We consequently call the radio emission
``non-thermal.''

\ni The distance to this brown dwarf is $d = 5$ pc.
Take the radio-emitting region to be a shell
of radius $R = 5 \times 10^{9}\cm$ (equal to the photospheric
radius of the brown dwarf) and thickness $\Delta R < R$.

\ni (a) Verify that the radio emission is indeed non-thermal.

\ni (b) Berger et al.~claim that the flux at $\nu_1$
is ``self-absorbed.'' Explain the justification behind
this statement.

\ni (c) Assume for the rest of this problem that
the flux at $\nu_1$ arises from optically thick
material, while the flux at $\nu_2$ arises from
marginally optically thin material. By ``marginally
optically thin,'' we mean that expressions for
synchrotron emission in both the optically thick
and thin limits will be approximately valid at $\nu_2$.

\ni Estimate the strength of the magnetic field ($B$ in Gauss),
the absolute number ($N_2$) of relativistic electrons
responsible for emission at $\nu_2$, and
the Lorentz factor ($\gamma_2$) of those electrons.

\ni Notice we are NOT referring explicitly to any
power-law distribution of electron energies. We cannot
talk about such distributions because we do not
have information on the optically thin portion of the spectrum.

\ni Work symbolically as much as possible and plug in
numbers only at the absolute end of the calculation.

\ni [It may interest you to know that Jupiter (which is less
massive than brown dwarfs) has a surface magnetic field of about 10 G.
Active M dwarfs (which are more massive than brown
dwarfs) have fields of several kG.]

\ni (d) Verify our assumption in (c) that
the flux at $\nu_2$ arises from (marginally) optically
thin material. That is, estimate the optical
depth across $\Delta R$ at $\nu_2$.



